Lesson 6: Chapter Summary and Review

New Concept

Linear models describe relationships with a constant rate of change. We represent them with tables, graphs, and key equation forms: slope-intercept ($y = b+mx$), point-slope ($y = y_1+m(x-x_1)$), and general form ($Ax+By=C$).

What’s next

Now, let's apply these formulas. Get ready for practice cards and challenge problems to master graphing and solving linear equations from various real-world scenarios.

Property

Linear models have equations of the form:

Examples

It costs 2000 dollars to develop a calculator and 20 dollars to manufacture each one. The total cost, $C$, for $n$ calculators is $C = 2000 + 20n$.

The world's oil reserves were 2100 billion barrels and decrease by 28 billion barrels per year. The remaining reserves, $R$, after $t$ years is $R = 2100 - 28t$.

Property

The general form for a linear equation is: $Ax + By = C$.

Examples

For $2x - 4y = 5$, if $x=0$, then $-4y=5$, so the y-intercept is $(0, -\frac{5}{4})$. If $y=0$, then $2x=5$, so the x-intercept is $(\frac{5}{2}, 0)$.

A vacation costs 60 dollars per day in one city ($C$) and 100 dollars per day in another ($T$), with a 1200 dollar budget. The equation is $60C + 100T = 1200$.

Property

The slope of a line gives us the rate of change of one variable with respect to another.

Formula for slope:

Examples

Find the slope between $(-1, 4)$ and $(3, -2)$: $m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}$.

Property

The slope-intercept form is $y = b + mx$. This form is useful when we know the initial value ($b$) and the rate of change ($m$).

Examples

For the line $2x - 4y = 5$, we can rewrite it as $-4y = 5 - 2x$, so $y = -\frac{5}{4} + \frac{1}{2}x$. The slope is $\frac{1}{2}$ and the y-intercept is $-\frac{5}{4}$.

An equation for temperature $T$ at altitude $h$ is $T = 62 - 0.0036h$. The slope is $-0.0036$ and the y-intercept (ground temperature) is 62.

Property

The point-slope form is $y = y_1 + m(x - x_1)$. This form is useful when we know the rate of change ($m$) and one point $(x_1, y_1)$ on the line.

Examples

Find the equation for a line with slope $m = -\frac{2}{3}$ passing through $(-4, 6)$. The equation is $y = 6 - \frac{2}{3}(x - (-4))$, which is $y = 6 - \frac{2}{3}(x + 4)$.

A line passes through $(3, -5)$ and $(-2, 4)$. The slope is $m = \frac{4 - (-5)}{-2 - 3} = -\frac{9}{5}$. Using point $(-2, 4)$, the equation is $y = 4 - \frac{9}{5}(x + 2)$.